Marginal quantile regression for dependent data with a working odds-ratio matrix

Marginal quantile regression for dependent data with a working odds-ratio matrix SUMMARY Dependent data arise frequently in applied research and several approaches to adjusting for the dependence among observations have been proposed in quantile regression. Cluster bootstrap is generally inefficient and computationally demanding, especially when the number of clusters is large. When the primary interest is on marginal quantiles, estimating equations have been proposed that estimate a working correlation matrix from the regression residuals’ sign. However, the Pearson’s correlation coefficient is an inadequate measure of dependence between binary variables because its range depends on their marginal probabilities. Instead, we propose to model the working correlation matrix through odds ratios. Different working structures can be easily estimated by suitable logistic regression models. These structures can be parametrized to depend on covariates and clusters. Simulations show that the proposed estimator has similar behavior to that of generalized estimating equations applied to regression for the mean. We study marginal quantiles of cognitive behavior with data from a randomized trial for treatment of obsessive compulsive disorder. 1. Introduction Longitudinal and clustered data represent two frequent data structures in which observations within clusters may be dependent. In this article, we analyze data from a longitudinal randomized trial on cognitive behavior therapy for treatment of obsessive compulsive disorder (Vigerland and others, 2016). Participants were randomized to 10 weeks of internet-based cognitive therapy with therapist support, or to a waitlist control condition. The main interest of the study was to assess the relationship between the treatment and anxiety disorder score. Several methods have been proposed to account for the dependence induced by the clustering when estimating marginal quantiles of a response variable. Cluster bootstrap is practical but can become computationally slow when the number of clusters is large. Besides, because it does not model the dependence, it may be inefficient. Generalized estimating equations (GEE), which can estimate population-averaged models, have become a popular alternative to the bootstrap. The dependence between observations within the same cluster is modeled through a correlation matrix, which is usually assumed to be the same for all clusters. In the literature, this matrix is called working correlation matrix, because the estimator is consistent even if the correlation is misspecified. The term “population-averaged” refers to the fact that the method models the average response over the subpopulation that shares a common value of the predictors as a function of such predictors (Diggle and others, 2002). Quantile regression is a distribution-free method (Koenker, 2005) that describes the entire conditional distribution of a response variable. Marginal quantiles were analyzed by Jung (1996), who linked the GEE approach to quantile regression. This method requires the estimation of the density of the residual errors, and is based on estimating equations that are non-smooth with respect to the parameters. More recently, Fu and Wang (2012) provided a smoothed version of Jung’s estimating equation by means of induced smoothing (Brown and Wang, 2005). This method does not require specifying the distribution of the residuals and estimates the parameters and their standard errors jointly. The main difference between mean and quantile GEE is related to the estimation of the working correlation matrix. In the quantile approach, this matrix is estimated from the regression residuals’ signs. However, the Pearson’s correlation coefficient is not a good measure of dependence between binary variables because it is bounded by their marginal probabilities. Because correlation is not a natural scale for binary sign variables, modeling on this scale has several disadvantages. First, it does not provide enough flexibility. For instance, parameters cannot depend on covariates. Second, estimation procedures might be complicated because they have to respect the constraints given by the marginal probabilities. Finally, the interpretation of the correlation coefficients may not be straightforward. Fu and others (2015) proposed to estimate the working correlation matrix using a Gaussian pseudolikelihood. Although this formulation may improve the flexibility, the computational and interpretation issues persist. Instead, we propose to model a working association matrix defined through odds ratios. Our approach is akin to two methods developed in the context of marginal logistic regression: alternating logistic regression initially proposed by Carey and others (1993) and the odds-ratio parameterization of binary-data association proposed by Lipsitz and others (1991). Touloumis and others (2013) studied the multinomial case. This approach was mentioned by Yi and He (2009) in the context of marginal quantile regression. This article explores the odds ratio parametrization in GEE applied to quantile regression. Different working structures can be estimated by specifying appropriate logistic regression models, including multilevel hierarchical data structures. Logistic regression models can be used to select the working dependence structure appropriately. The rest of the article is organized as follows. Section 2 presents the smoothed quantile GEE and the odds-ratio models. A simulation study is described in Section 3. We analyze data from a randomized trial on cognitive behavior therapy for treatment of obsessive compulsive disorder in Section 4. The article is concluded with a discussion in Section 5. 2. Methods Let $$\{y_{ij}, x_{ij} \}$$, $$i=1,\dots,n$$, $$j=1,\dots, T_i$$ be longitudinal data, where $$y_{ij} \in \mathbb{R}$$ is the response variable and $$x_{ij} \in \mathbb{R}^P$$ is the covariate vector. For brevity, we assume that the number of observations in a cluster $$T_i$$ is constant across clusters, $$T_i=T$$. The general case when $$T_i $$ varies across clusters is a straightforward extension. Consider the problem of estimating the conditional $$\tau$$-th quantile of $$y$$ given $$x$$. A simple solution is to treat the observation as independent and minimize the following objective function: \begin{equation*} \sum_{i,j} \epsilon_{ij} \; \psi (\epsilon_{ij}), \end{equation*} where $$\epsilon_{ij}=y_{ij}-x_{ij}^\top\beta_{\tau}$$ indicates the residual and $$\psi(\epsilon_{ij})=\tau - I(\epsilon_{ij}<0)$$ is a linear transformation of the residual sign. The parameter vector $$\beta_{\tau}$$ can be estimated by solving \begin{equation*} \sum_{i,j} x_{ij} \psi(\epsilon_{ij})=0 \end{equation*} Ignoring the dependence within clusters may lead to wrong standard errors. Consider the set of repeated measures on the $$i$$-th individual, denoted by $$y_i=(y_{i1}, \dots, y_{iT})$$, and its design matrix $$x_i=(x_{i1}, \dots, x_{iT})$$. Each element of the vector $$\psi(\epsilon_i)=(\psi(\epsilon_{i1}),\dots,\psi(\epsilon_{iT}))$$ follows a Bernoulli distribution with expectation $$\tau$$. Therefore, marginal quantile regression can be regarded as a special case of GEE where the mean model is a constant $$\tau$$ and the response variable contains a function of the parameters, $$I(\epsilon_{ij}<0)$$. Jung (1996) showed that marginal quantiles can be obtained by \begin{equation} U_Q(\beta)=\sum_{i=1}^{n} x_i^\top \Gamma_i W_i^{-1} \psi_{\tau}(\epsilon_{i})=0 \end{equation} (2.1) where $$W_i=\text{Cov} (\psi_{\tau}(y_i-x_i^\top \beta_{\tau}))$$ is the residuals sign covariance matrix of the $$i$$-th individual, $$\Gamma_i = \textrm{diag} (f_{i1}(0)), \dots,f_{i_T}(0))$$, and $$f_{ij}$$ indicates the probability density function of $$\epsilon_{ij}$$. The latter can be estimated by (Hall and Sheather, 1988) \begin{equation*} \hat{f}_{ij}(0)=2 h_n \left[x_{ij}^\top \left\{ \hat{\beta}_{\tau + h_n} - \hat{\beta}_{\tau - h_n} \right\}\right]^{-1} \end{equation*} where $$h_n$$ is a bandwidth parameter such that $$h_n \to 0$$ for $$n \to \infty$$, often calculated as $$h_n=1.57 n^{-1/3} \left( 1.5 \phi^2 \left\{ \Phi^{-1} (\tau) \right\} / \left[ 2 \left\{ \Phi^{-1} (\tau)\right\}^2 \right] \right)^{2/3}$$, where $$\phi$$ and $$\Phi$$ indicate the standard normal density and cumulative distribution, respectively. The covariance matrix $$W_i$$ can be parametrized to increase efficiency. To protect against misspecification, a sandwich estimator of the standard errors can be used. The correlation matrix of the regression residuals and of the regression residuals’ sign can be different from each other. As pointed out by Leng and Zhang (2014), if $$\epsilon_i$$ has an AR(1) covariance matrix with parameter $$\phi$$ then $$\psi(\epsilon_i)$$ depends on $$\phi$$ as a function of a computationally intractable 2D integral. Other examples can be found in Fu and others (2015). In general, the regression residuals and their signs have the same correlation structure only if $$\epsilon_i$$ has an exchangeable or Toeplitz covariance structure. Fu and Wang (2012) proposed to smooth Jung’s estimating equation by means of induced smoothing (Brown and Wang, 2005). They approximated the estimator by adding to the true value $$\beta$$ a multivariate standard normal distribution $$Z$$ and a smoothing parameter $$\Omega$$, $$\hat{\beta}=\beta + \Omega^{1/2} Z$$. The smoothed estimating equations are obtained by \begin{equation} \tilde{U}_Q(\beta)=E_Z(U_Q(\beta + \Omega^{1/2} Z))=\sum_{i=1}^{n} x_i^\top \Gamma_i W_i^{-1} (\eta) \tilde{\psi}_{\tau}(\epsilon_{i})=0 \end{equation} (2.2) where $$\tilde{\psi}_{\tau}(\epsilon_{i})=\left( 1- \Phi \left(\frac{y_{i1}-x_{i1}^\top \beta}{r_{i1}}\right),\dots, 1- \Phi \left(\frac{y_{iT}-x_{iT}^\top \beta}{r_{iT}}\right) \right)^\top $$, $$\Phi(\cdot)$$ is the standard normal cumulative distribution and $$r_{ik}=(x_{ik}^\top \Omega x_{ik})^{1/2}$$. The derivatives of the smoothed score are \begin{equation*} \tilde{D}(\beta)=\frac{\partial \tilde{U}_Q(\beta)}{\partial \beta}=\sum_{i=1}^n X_i^\top \Gamma_i W_i^{-1} (\eta) \tilde{\Lambda}_i X_i, \end{equation*} where $$\tilde{\Lambda}_i$$ is a diagonal matrix with $$k$$th diagonal element $$r_{ik}^{-1} \phi ((y_{ik}-x_{ik}^\top \beta)/r_{ik})$$. The estimation of $$\tilde{\beta}$$ and its covariance matrix is obtained through an algorithm, summarized by the following steps: 1. Initialization: obtain $$\tilde{\beta}^0$$ using ordinary quantile regression; set $$\Omega^0=n^{-1}I_p$$ and $$K=0$$. 2. Compute the covariance matrix $$W_i(\eta)$$ using logistic regression on the residuals sign of the current estimate $$\tilde{\beta^K}$$. 3. Update $$\tilde{\beta}^{K+1}$$ and $$\tilde{\Omega}^{K+1}$$ by: \begin{eqnarray*} \tilde{\beta}^{K+1}&=& {\beta}^{K} + \left\{ - \tilde{D}(\tilde{\beta}^{K}, \tilde{\Omega}^{K} ) \right\}^{-1} \tilde{U}_Q\left(W_i^{K},\tilde{\beta}^{K}, \tilde{\Omega}^{K}\right) \\ \tilde{\Omega}^{K+1}&=&\left[\tilde{D}\left(\tilde{\beta}^{K+1}, \tilde{\Omega}^{K}\right)\right]^{-1} \textrm{Cov}\left\{\tilde{U}_Q\Big(W_i^{K},\tilde{\beta}^{K+1}\right\} \left\{\left[\tilde{D}\left(\tilde{\beta}^{K+1}, \tilde{\Omega}^{K}\right)\right]^{-1}\right\}^{T}, \end{eqnarray*} where $$ \textrm{Cov}\{\tilde{U}_Q(\beta)\} = \sum_{i=1}^n x_i^\top \Gamma_i W_i^{-1} (\eta) \tilde{\psi}_{\tau}(\epsilon_i) \tilde{\psi}^\top_{\tau}(\epsilon_i) W_i^{-1}(\eta) \Gamma_i x_i.$$ 4. Repeat steps (2) and (3) until convergence. The final values of $$\tilde{\beta}$$ and $$\tilde{\Omega}$$ are the smoothed estimator of $$\beta$$ and its covariance matrix. Fu and Wang (2012) discussed the regularity conditions under which the estimators derived by the above smoothing method are consistent and asymptotically normally distributed. The algorithm is fast and it usually requires few iterations to achieve convergence. Some difficulties may arise when estimating marginal quantiles at extreme quantiles, because $$W_i(\eta)$$ is less likely to be positive definite. 2.1 Estimating the working association matrix Let $$V_{it} = I (y_{it} \le x_{it}^\top \beta_{\tau})$$ be the residual sign of the $$i$$-th individual at time $$t$$ and $$V_t=(V_{1t},\dots,V_{nt})$$ be the set of residual signs at time $$t$$. A generic element $$w_{zu}$$ of the working covariance matrix $$W_i(\eta)$$ can be written as \begin{equation*} w_{zu}= \begin{cases} \tau (1-\tau), & z=u \\ p_{zu} - \tau^2, & z \ne u \\ \end{cases} \end{equation*} where $$p_{zu}=E(V_{z} V_{u})=P(V_{z}=1,V_{u}=1)$$. The elements of $$W_i(\eta)$$ can be computed simultaneously through a second set of estimating equations (Prentice, 1988). These probabilities are bounded by $$\tau$$, the marginal probability of $$V_{z}$$ and $$V_{u}$$, as follows: \begin{equation} \text{max}(0,2 \tau -1) \le p_{zu} \le \tau. \end{equation} (2.3) The correlation coefficient is not a good measure of dependence between binary variables, because it is bounded by their marginal frequencies. Instead, we propose to model a working association matrix defined through odds ratios. Let $$\eta_{zu}$$ be the odds ratio between $$V_{z}$$ and $$V_{u}$$, \begin{equation*} \eta_{zu}=\frac{P(V_{z}=1,V_{u}=1) / P(V_{z}=0,V_{u}=0)}{P(V_{z}=0,V_{u}=1) / P(V_{z}=1,V_{u}=0)}. \end{equation*} Let $$\mathcal{A}=\{(V_z,V_u)\},\;z=u,\dots,T,\; u=1,\dots,T,$$ be the set of pairwise vectors of all $$\dbinom{T}{2}$$ residual signs corresponding to the odds ratios in the lower triangular part of the working correlation matrix $$W_i(\eta)$$. Because $$W_i(\eta)$$ is symmetric, the upper triangular part is equal to the lower part. Consider the new data set $$(V_z, V_u , z, u)$$. For any working structure of $$W_i (\eta)$$, the respective set of odds ratios can be estimated simultaneously through an appropriate definition of the linear predictor in a logistic regression of $$V_z$$ on $$V_u$$, $$V_z|V_u \sim Be(\mu_{zu})$$. Estimating $$W_i(\eta)$$ using logistic models has some advantages. First, it makes it easy to specify the form of the working correlation matrix. For example, Exchangeable: $$\textrm{logit}(\mu_{zu})=\alpha + \eta V_u$$; Toeplitz: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{T-1} \eta_{i} I_{z-u=i} V_u + \sum_{i=1}^{T-1} I_{z-u=i}$$; Unstructured: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{{T}\choose{2}} \eta_{i} I_{c=i} V_u + \sum_{i=1}^{{T}\choose{2}}I_{c=i}$$. Nested Exchangeable: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{K-1} \eta_i V_u I_{c_i=1} + \sum_{i=1}^{K-1} I_{c_i=1}$$, where $$c_i$$ indicates whether $$V_z$$ and $$V_u$$ belong to the same $$i$$-th cluster. Exchangeable varying with a covariate $$X$$: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{K} \eta_i V_u I_{\textit{X}=i} + \sum_{i=1}^{1} I_{\textit{X}=i}$$, where $$K$$ indicates the number of categories of $$X$$. Second, logistic models can be used to easily select the most appropriate structure of the working correlation matrix. Likelihood comparisons and Akaike’s information criterion may help identify the best structure. Given the marginal probabilities $$\tau$$ and the odds ratios $$\eta_{zu}$$, the joint probabilities $$p_{zu}$$ can be obtained by solving the following equation: \begin{equation*} (\eta_{zu} -1) p_{zu}^2 + ( 2 \tau (1 - \eta_{zu}) -1)p_{zu} + \tau^2 \eta_{zu}=0. \end{equation*} Only the smaller root of the previous equation provides probabilities that respect the constraints in (2.3). 3. Simulation study We conducted simulations to assess the performance of the proposed method. The response variable was generated from the following model: \begin{equation*} y_{ij}=\beta_0 +\beta_1 x_{1ij} + (1+ |x_{1ij}|) (\epsilon_{ij} - q_{\tau}) \;\;\; i=1,\dots,250,\;\;j=1,\dots,T, \end{equation*} where $$q_{\tau}$$ was such that $$p(\epsilon_{ij} \le q_{\tau}) = \tau$$. The size of each cluster was sampled from a binomial distribution $$\textrm{Binom}(T, p)$$, with $$T = (5,10)$$ and $$p = 0.5$$. The covariate $$x_{1ij}$$ was sampled from a uniform distribution $$U(0,1)$$ and the regression coefficients were set to one. The distribution of the error term, $$\epsilon_{i}=(\epsilon_{i1},\dots,\epsilon_{iT})^\top$$, was multivariate Gaussian with variance equal to 10. We considered the following dependence structures: Exchangeable with correlation $$\rho = 0.3$$ and $$\rho = 0.6$$; Independent; Toeplitz with correlation $$\rho = 0.4$$ for the first two lags and zero otherwise. We conducted a simulation study with 500 independent realizations and estimated three different quantiles, $$\tau=(0.10, 0.25, 0.50)$$. All computations were performed using R version 3.23. The programming code is available from the author upon request. The multivariate Gaussian random variables were generated using the “mvtnorm” library. Model-based (naive) standard errors were on average smaller than robust standard errors in all settings (Table 1), except the independence case. Misspecification of the working correlation structure lead to a decrease in the empirical coverage of the estimators. Coverage probabilities of the robust standard errors were close to their nominal value in all the settings (Table 2). The estimator of the 10-th percentile showed undercoverage, with smallest observed value equal to 0.88. The average standard errors of the estimators were near those observed over the Monte Carlo replicates across all settings (Tables 3 and 4). Cluster bootstrap had a smaller mean squared error (MSE) and average standard error than the proposed method when the 10-th percentile was estimated (Table 5). When the 25-th and 50-th percentile were estimated, the proposed method provided smaller MSE and average standard error than cluster bootstrap, regardless of the dependence structure. Similarly to the classic GEE for the expectation of a response variable, selecting the true dependence structure improved the performances of the estimator. When the true dependence structure was Toeplitz, the estimator obtained using the true dependence structure provided the smallest MSE and average standard errors in most of the settings. In the independent case, results were very similar across all the working structures. In the exchangeable case, the independence and exchangeable structures had the smallest MSE and average standard errors. We computed also the bias of the estimator, which is shown in Table 6. Table 1 Standard error difference between the robust and naive estimators $$($$Avg s.e.$$)$$ and empirical coverage of the naive estimators $$($$Cover$$)$$ for the parameter $$\beta_1$$ at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; TO, Toeplitz $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 Table 1 Standard error difference between the robust and naive estimators $$($$Avg s.e.$$)$$ and empirical coverage of the naive estimators $$($$Cover$$)$$ for the parameter $$\beta_1$$ at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; TO, Toeplitz $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 Table 2 Empirical coverage of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 Table 2 Empirical coverage of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 Table 3 Average standard error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 Table 3 Average standard error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 Table 4 Monte Carlo standard deviation of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 Table 4 Monte Carlo standard deviation of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 Table 5 Mean squared error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 Table 5 Mean squared error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 Table 6 Bias of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 Table 6 Bias of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 Model selection performed through Akaike’s information criterion selected the true dependence structure with high probability (Table 7). The smallest percentage of correct selection was 40%, obtained when the true dependence structure was Toeplitz, $$\tau=0.10$$ and $$T=5$$. The highest was 100%, obtained when the true dependence structure was Toeplitz, $$\tau=0.50$$ and $$T=10$$. In general, the percentage of correct selection was higher for the median than for the 10-th and 25-th percentiles. Table 7 Observed percentage of correct selections of the true model in different scenarios and quantiles. $$T$$ indicates the cluster size. WI, independence; EX, exchangeable; TO, Toeplitz; AR, autoregressive $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 Table 7 Observed percentage of correct selections of the true model in different scenarios and quantiles. $$T$$ indicates the cluster size. WI, independence; EX, exchangeable; TO, Toeplitz; AR, autoregressive $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 We also explored the effect of cluster size by raising the binomial probability in the data-generating model to $$p = 0.8$$. The results were nearly unchanged and are not shown. We also considered a sample size of $$n = 100$$ and $$T = 15$$. The properties of the estimators were similar to those obtained with the previous settings. (The results are not shown.) 4. Application to internet-based cognitive therapy data set The data set consisted of 95 ($$n=95$$) families with a child aged 8–12 years with a principal diagnosis of generalized anxiety, panic disorder, separation anxiety, social phobia, and specific phobia. Participants were randomized to ten weeks of internet-based cognitive therapy with therapist support ($$70$$ families), or to a waitlist control condition ($$25$$ families). At weekly intervals, the amount of difficulties in emotion regulation scale (DERS) was measured on a 0–100 scale, where 0 indicated no difficulties and 100 extreme difficulties. The maximum number of repeated measurements is fourteen ($$T=14$$). Similarly to previous studies, we assumed that data were missing completely at random. The main interest of the study was to assess the relationship between the treatment and DERS. Figure 1 shows the residual errors of a classical mean regression and their kernel density. It was markedly skewed and there were several outliers. The mean was not an appropriate summary of it, and we considered quantiles. There seemed to be a time effect on the response variable, both in treated and untreated groups (Figure 2). The intraclass correlation was estimated to be 70%. Fig. 1. View largeDownload slide Mean regression residuals obtained from model on the right-hand side of the equation (4.4) and their estimated kernel density. Fig. 1. View largeDownload slide Mean regression residuals obtained from model on the right-hand side of the equation (4.4) and their estimated kernel density. Fig. 2. View largeDownload slide Boxplot of DERS at different times in treated (top panel) and untreated (bottom panel) children. In the treated group the sample sizes at the 14 time points were, respectively: 58, 52, 51, 41, 49, 39, 43, 39, 43, 35, 27, 32, 32, and 30 children. In the untreated group the sample sizes at the 14 time points were, respectively: 19, 20, 16, 16, 13, 14, 13, 12, 13, 9, 12, 12, 12, and 12 children. Fig. 2. View largeDownload slide Boxplot of DERS at different times in treated (top panel) and untreated (bottom panel) children. In the treated group the sample sizes at the 14 time points were, respectively: 58, 52, 51, 41, 49, 39, 43, 39, 43, 35, 27, 32, 32, and 30 children. In the untreated group the sample sizes at the 14 time points were, respectively: 19, 20, 16, 16, 13, 14, 13, 12, 13, 9, 12, 12, 12, and 12 children. We considered three different quantiles, $$\tau=(0.25, 0.50, 0.75)$$, and estimated the following model: \begin{equation} Q(\textrm{Ders}_{ij} | \textrm{Treat}, \textrm{Time})=\beta_0 +\beta_1 \textrm{Treat}_{ij} + \beta_2 \textrm{Time}_{ij} + \beta_3 \textrm{Treat}_{ij} \times \textrm{Time}_{ij}. \end{equation} (4.4) We considered an exchangeable, Toeplitz, independence, and exchangeable varying with treatment correlation structures, along with a cluster bootstrap. Results are shown in Table 8. Table 8 Estimated quantile regression coefficients, standard errors $$($$in parentheses$$)$$ and logistic models’ AIC for three quantiles using exchangeable $$($$EX$$)$$, exchangeable varying with treatment $$($$EX-Trt$$)$$, Toeplitz $$($$TO$$)$$, and independence $$($$WI$$)$$ working correlation structures, along with a cluster bootstrap $$($$BT$$)$$ Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) Table 8 Estimated quantile regression coefficients, standard errors $$($$in parentheses$$)$$ and logistic models’ AIC for three quantiles using exchangeable $$($$EX$$)$$, exchangeable varying with treatment $$($$EX-Trt$$)$$, Toeplitz $$($$TO$$)$$, and independence $$($$WI$$)$$ working correlation structures, along with a cluster bootstrap $$($$BT$$)$$ Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) We found a significant effect of treatment at the $$50$$-th and $$25$$-th percentiles in the exchangeable and exchangeable varying with treatment correlation structures. In the Toeplitz structure, this effect was significant only at the $$25$$-th percentile. The coefficient for time was significant at the median in all correlation structures except the Toeplitz. It was significant at the $$75$$-th percentile in all structures. The exchangeable structure varying with treatment had the smallest Akaike’s information criterion. Measurements from subjects that were treated were less correlated than those from subjects that were not treated (Table 9). Standard errors obtained with cluster bootstrap were up to 50% greater than those obtained with the proposed method. Table 9 Estimated working correlation parameters for the exchangeable ($$\phi_{\textrm{Exch}}$$) and exchangeable varying with treatment ($$\phi_{\textrm{TRT=1}}$$ and $$\phi_{\textrm{TRT=0}}$$) structures Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Table 9 Estimated working correlation parameters for the exchangeable ($$\phi_{\textrm{Exch}}$$) and exchangeable varying with treatment ($$\phi_{\textrm{TRT=1}}$$ and $$\phi_{\textrm{TRT=0}}$$) structures Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Figure 3 shows the relationship between DERS and time, stratified by treatment. As the estimated quantile increases, the correlation between DERS and time decreases. The differences between treated and untreated are stronger at the median than at other percentiles. Fig. 3. View largeDownload slide Quantile regression for DERS as a function of time by treatment at three different quantiles $$\tau=(0.25,0.50,0.75)$$ with an exchangeable correlation structure varying with treatment. The dashed lines represent the treated group, while the solid lines represent the untreated group. Fig. 3. View largeDownload slide Quantile regression for DERS as a function of time by treatment at three different quantiles $$\tau=(0.25,0.50,0.75)$$ with an exchangeable correlation structure varying with treatment. The dashed lines represent the treated group, while the solid lines represent the untreated group. Because the interaction term was not significant in all percentiles, we concluded that the treatment did not significantly decrease the anxiety disorder of children. 5. Discussion This article describes an alternative method to estimate the dependence within clusters for inference on population marginal quantile models. The association matrix of the residuals’ sign is estimated by means of appropriate logistic regression models. The estimation is flexible and computationally fast. In our study, the proposed method was approximately as fast as GEE applied to the mean. At extreme quantiles, the algorithm occasionally failed to converge. This partly accounts for the observed larger MSE of the proposed method with respect to the bootstrap at the 10-percentile in our simulation study. The best-fitting working structure can be selected by likelihood comparisons or, for non-nested models, by the Akaike’s information criterion. Our simulations showed that this criterion could select the true dependence structure with high probability. In other settings, however, other criteria may be preferable (Jaman and others, 2016). In the scenarios considered in our simulation, the asymptotic approximations for the covariance matrix of the estimators were satisfactory. When the asymptotic approximations are suspected to be poor, one may consider running a simulation specifically tailored to the data at hand. Possible extensions of this work include models for dependence with non-parametric quantile regression, M-estimators, LMS (Cole, 1990), and other classifiers. When missing data are present, and the non-response mechanism is suspected to be missing at random, weighted estimating equations can be applied. Yi and He (2009) discussed the use of weighted estimating equations for median regression for longitudinal data with dropouts, and demonstrated the consistency and asymptotic distribution of the resulting estimators. Acknowledgments onflict of Interest: None declared. References Brown B. M. and Wang Y.-G. ( 2005 ). Standard errors and covariance matrices for smoothed rank estimators. 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Google Scholar Crossref Search ADS Hall P. and Sheather S. J. ( 1988 ). On the distribution of a studentized quantile. Journal of the Royal Statistical Society Series B (Methodological) , 50 , 381 – 391 . Jaman A. , Latif, Mahbub A. H. M. , Barib W. and Wahedc A. S. ( 2016 ). A determinant-based criterion for working correlation structure selection in generalized estimating equations. Statistics in Medicine 35 , 1819 – 1833 . Google Scholar Crossref Search ADS PubMed Jung S. ( 1996 ). Quasi-likelihood for median regression models. Journal of the American Statistical Association 433 , 251 – 257 . Google Scholar Crossref Search ADS Koenker R. ( 2005 ). Quantile Regression , 38 . Cambridge : Cambridge University Press. Leng C. and Zhang W. ( 2014 ). Smoothing combined estimating equations in quantile regression for longitudinal data. Statistics and Computing 24 , 123 – 136 . Google Scholar Crossref Search ADS Lipsitz S. R. , Laird N. M. and Harrington D. P. ( 1991 ). Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association. Biometrika 78 , 153 – 160 . Google Scholar Crossref Search ADS Prentice R. L. ( 1988 ). Correlated binary regression with covariates specific to each binary observation. Biometrics , 1033 – 1048 . Touloumis A. , Agresti A. and Kateri M. ( 2013 ). Gee for multinomial responses using a local odds ratios parameterization. Biometrics 69 , 633 – 640 . Google Scholar Crossref Search ADS PubMed Vigerland S. , Ljótsson B. , Thulin U. , öst L. -G. , Andersson G. and Serlachius E. ( 2016 ). Internet-delivered cognitive behavioural therapy for children with anxiety disorders: A randomised controlled trial. Behaviour Research and Therapy 76 , 47 – 56 . Google Scholar Crossref Search ADS PubMed Yi G. Y. and He W. ( 2009 ). Median regression models for longitudinal data with dropouts. Biometrics 65 , 618 – 625 . Google Scholar Crossref Search ADS PubMed © The Author 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biostatistics Oxford University Press

Marginal quantile regression for dependent data with a working odds-ratio matrix

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Abstract

SUMMARY Dependent data arise frequently in applied research and several approaches to adjusting for the dependence among observations have been proposed in quantile regression. Cluster bootstrap is generally inefficient and computationally demanding, especially when the number of clusters is large. When the primary interest is on marginal quantiles, estimating equations have been proposed that estimate a working correlation matrix from the regression residuals’ sign. However, the Pearson’s correlation coefficient is an inadequate measure of dependence between binary variables because its range depends on their marginal probabilities. Instead, we propose to model the working correlation matrix through odds ratios. Different working structures can be easily estimated by suitable logistic regression models. These structures can be parametrized to depend on covariates and clusters. Simulations show that the proposed estimator has similar behavior to that of generalized estimating equations applied to regression for the mean. We study marginal quantiles of cognitive behavior with data from a randomized trial for treatment of obsessive compulsive disorder. 1. Introduction Longitudinal and clustered data represent two frequent data structures in which observations within clusters may be dependent. In this article, we analyze data from a longitudinal randomized trial on cognitive behavior therapy for treatment of obsessive compulsive disorder (Vigerland and others, 2016). Participants were randomized to 10 weeks of internet-based cognitive therapy with therapist support, or to a waitlist control condition. The main interest of the study was to assess the relationship between the treatment and anxiety disorder score. Several methods have been proposed to account for the dependence induced by the clustering when estimating marginal quantiles of a response variable. Cluster bootstrap is practical but can become computationally slow when the number of clusters is large. Besides, because it does not model the dependence, it may be inefficient. Generalized estimating equations (GEE), which can estimate population-averaged models, have become a popular alternative to the bootstrap. The dependence between observations within the same cluster is modeled through a correlation matrix, which is usually assumed to be the same for all clusters. In the literature, this matrix is called working correlation matrix, because the estimator is consistent even if the correlation is misspecified. The term “population-averaged” refers to the fact that the method models the average response over the subpopulation that shares a common value of the predictors as a function of such predictors (Diggle and others, 2002). Quantile regression is a distribution-free method (Koenker, 2005) that describes the entire conditional distribution of a response variable. Marginal quantiles were analyzed by Jung (1996), who linked the GEE approach to quantile regression. This method requires the estimation of the density of the residual errors, and is based on estimating equations that are non-smooth with respect to the parameters. More recently, Fu and Wang (2012) provided a smoothed version of Jung’s estimating equation by means of induced smoothing (Brown and Wang, 2005). This method does not require specifying the distribution of the residuals and estimates the parameters and their standard errors jointly. The main difference between mean and quantile GEE is related to the estimation of the working correlation matrix. In the quantile approach, this matrix is estimated from the regression residuals’ signs. However, the Pearson’s correlation coefficient is not a good measure of dependence between binary variables because it is bounded by their marginal probabilities. Because correlation is not a natural scale for binary sign variables, modeling on this scale has several disadvantages. First, it does not provide enough flexibility. For instance, parameters cannot depend on covariates. Second, estimation procedures might be complicated because they have to respect the constraints given by the marginal probabilities. Finally, the interpretation of the correlation coefficients may not be straightforward. Fu and others (2015) proposed to estimate the working correlation matrix using a Gaussian pseudolikelihood. Although this formulation may improve the flexibility, the computational and interpretation issues persist. Instead, we propose to model a working association matrix defined through odds ratios. Our approach is akin to two methods developed in the context of marginal logistic regression: alternating logistic regression initially proposed by Carey and others (1993) and the odds-ratio parameterization of binary-data association proposed by Lipsitz and others (1991). Touloumis and others (2013) studied the multinomial case. This approach was mentioned by Yi and He (2009) in the context of marginal quantile regression. This article explores the odds ratio parametrization in GEE applied to quantile regression. Different working structures can be estimated by specifying appropriate logistic regression models, including multilevel hierarchical data structures. Logistic regression models can be used to select the working dependence structure appropriately. The rest of the article is organized as follows. Section 2 presents the smoothed quantile GEE and the odds-ratio models. A simulation study is described in Section 3. We analyze data from a randomized trial on cognitive behavior therapy for treatment of obsessive compulsive disorder in Section 4. The article is concluded with a discussion in Section 5. 2. Methods Let $$\{y_{ij}, x_{ij} \}$$, $$i=1,\dots,n$$, $$j=1,\dots, T_i$$ be longitudinal data, where $$y_{ij} \in \mathbb{R}$$ is the response variable and $$x_{ij} \in \mathbb{R}^P$$ is the covariate vector. For brevity, we assume that the number of observations in a cluster $$T_i$$ is constant across clusters, $$T_i=T$$. The general case when $$T_i $$ varies across clusters is a straightforward extension. Consider the problem of estimating the conditional $$\tau$$-th quantile of $$y$$ given $$x$$. A simple solution is to treat the observation as independent and minimize the following objective function: \begin{equation*} \sum_{i,j} \epsilon_{ij} \; \psi (\epsilon_{ij}), \end{equation*} where $$\epsilon_{ij}=y_{ij}-x_{ij}^\top\beta_{\tau}$$ indicates the residual and $$\psi(\epsilon_{ij})=\tau - I(\epsilon_{ij}<0)$$ is a linear transformation of the residual sign. The parameter vector $$\beta_{\tau}$$ can be estimated by solving \begin{equation*} \sum_{i,j} x_{ij} \psi(\epsilon_{ij})=0 \end{equation*} Ignoring the dependence within clusters may lead to wrong standard errors. Consider the set of repeated measures on the $$i$$-th individual, denoted by $$y_i=(y_{i1}, \dots, y_{iT})$$, and its design matrix $$x_i=(x_{i1}, \dots, x_{iT})$$. Each element of the vector $$\psi(\epsilon_i)=(\psi(\epsilon_{i1}),\dots,\psi(\epsilon_{iT}))$$ follows a Bernoulli distribution with expectation $$\tau$$. Therefore, marginal quantile regression can be regarded as a special case of GEE where the mean model is a constant $$\tau$$ and the response variable contains a function of the parameters, $$I(\epsilon_{ij}<0)$$. Jung (1996) showed that marginal quantiles can be obtained by \begin{equation} U_Q(\beta)=\sum_{i=1}^{n} x_i^\top \Gamma_i W_i^{-1} \psi_{\tau}(\epsilon_{i})=0 \end{equation} (2.1) where $$W_i=\text{Cov} (\psi_{\tau}(y_i-x_i^\top \beta_{\tau}))$$ is the residuals sign covariance matrix of the $$i$$-th individual, $$\Gamma_i = \textrm{diag} (f_{i1}(0)), \dots,f_{i_T}(0))$$, and $$f_{ij}$$ indicates the probability density function of $$\epsilon_{ij}$$. The latter can be estimated by (Hall and Sheather, 1988) \begin{equation*} \hat{f}_{ij}(0)=2 h_n \left[x_{ij}^\top \left\{ \hat{\beta}_{\tau + h_n} - \hat{\beta}_{\tau - h_n} \right\}\right]^{-1} \end{equation*} where $$h_n$$ is a bandwidth parameter such that $$h_n \to 0$$ for $$n \to \infty$$, often calculated as $$h_n=1.57 n^{-1/3} \left( 1.5 \phi^2 \left\{ \Phi^{-1} (\tau) \right\} / \left[ 2 \left\{ \Phi^{-1} (\tau)\right\}^2 \right] \right)^{2/3}$$, where $$\phi$$ and $$\Phi$$ indicate the standard normal density and cumulative distribution, respectively. The covariance matrix $$W_i$$ can be parametrized to increase efficiency. To protect against misspecification, a sandwich estimator of the standard errors can be used. The correlation matrix of the regression residuals and of the regression residuals’ sign can be different from each other. As pointed out by Leng and Zhang (2014), if $$\epsilon_i$$ has an AR(1) covariance matrix with parameter $$\phi$$ then $$\psi(\epsilon_i)$$ depends on $$\phi$$ as a function of a computationally intractable 2D integral. Other examples can be found in Fu and others (2015). In general, the regression residuals and their signs have the same correlation structure only if $$\epsilon_i$$ has an exchangeable or Toeplitz covariance structure. Fu and Wang (2012) proposed to smooth Jung’s estimating equation by means of induced smoothing (Brown and Wang, 2005). They approximated the estimator by adding to the true value $$\beta$$ a multivariate standard normal distribution $$Z$$ and a smoothing parameter $$\Omega$$, $$\hat{\beta}=\beta + \Omega^{1/2} Z$$. The smoothed estimating equations are obtained by \begin{equation} \tilde{U}_Q(\beta)=E_Z(U_Q(\beta + \Omega^{1/2} Z))=\sum_{i=1}^{n} x_i^\top \Gamma_i W_i^{-1} (\eta) \tilde{\psi}_{\tau}(\epsilon_{i})=0 \end{equation} (2.2) where $$\tilde{\psi}_{\tau}(\epsilon_{i})=\left( 1- \Phi \left(\frac{y_{i1}-x_{i1}^\top \beta}{r_{i1}}\right),\dots, 1- \Phi \left(\frac{y_{iT}-x_{iT}^\top \beta}{r_{iT}}\right) \right)^\top $$, $$\Phi(\cdot)$$ is the standard normal cumulative distribution and $$r_{ik}=(x_{ik}^\top \Omega x_{ik})^{1/2}$$. The derivatives of the smoothed score are \begin{equation*} \tilde{D}(\beta)=\frac{\partial \tilde{U}_Q(\beta)}{\partial \beta}=\sum_{i=1}^n X_i^\top \Gamma_i W_i^{-1} (\eta) \tilde{\Lambda}_i X_i, \end{equation*} where $$\tilde{\Lambda}_i$$ is a diagonal matrix with $$k$$th diagonal element $$r_{ik}^{-1} \phi ((y_{ik}-x_{ik}^\top \beta)/r_{ik})$$. The estimation of $$\tilde{\beta}$$ and its covariance matrix is obtained through an algorithm, summarized by the following steps: 1. Initialization: obtain $$\tilde{\beta}^0$$ using ordinary quantile regression; set $$\Omega^0=n^{-1}I_p$$ and $$K=0$$. 2. Compute the covariance matrix $$W_i(\eta)$$ using logistic regression on the residuals sign of the current estimate $$\tilde{\beta^K}$$. 3. Update $$\tilde{\beta}^{K+1}$$ and $$\tilde{\Omega}^{K+1}$$ by: \begin{eqnarray*} \tilde{\beta}^{K+1}&=& {\beta}^{K} + \left\{ - \tilde{D}(\tilde{\beta}^{K}, \tilde{\Omega}^{K} ) \right\}^{-1} \tilde{U}_Q\left(W_i^{K},\tilde{\beta}^{K}, \tilde{\Omega}^{K}\right) \\ \tilde{\Omega}^{K+1}&=&\left[\tilde{D}\left(\tilde{\beta}^{K+1}, \tilde{\Omega}^{K}\right)\right]^{-1} \textrm{Cov}\left\{\tilde{U}_Q\Big(W_i^{K},\tilde{\beta}^{K+1}\right\} \left\{\left[\tilde{D}\left(\tilde{\beta}^{K+1}, \tilde{\Omega}^{K}\right)\right]^{-1}\right\}^{T}, \end{eqnarray*} where $$ \textrm{Cov}\{\tilde{U}_Q(\beta)\} = \sum_{i=1}^n x_i^\top \Gamma_i W_i^{-1} (\eta) \tilde{\psi}_{\tau}(\epsilon_i) \tilde{\psi}^\top_{\tau}(\epsilon_i) W_i^{-1}(\eta) \Gamma_i x_i.$$ 4. Repeat steps (2) and (3) until convergence. The final values of $$\tilde{\beta}$$ and $$\tilde{\Omega}$$ are the smoothed estimator of $$\beta$$ and its covariance matrix. Fu and Wang (2012) discussed the regularity conditions under which the estimators derived by the above smoothing method are consistent and asymptotically normally distributed. The algorithm is fast and it usually requires few iterations to achieve convergence. Some difficulties may arise when estimating marginal quantiles at extreme quantiles, because $$W_i(\eta)$$ is less likely to be positive definite. 2.1 Estimating the working association matrix Let $$V_{it} = I (y_{it} \le x_{it}^\top \beta_{\tau})$$ be the residual sign of the $$i$$-th individual at time $$t$$ and $$V_t=(V_{1t},\dots,V_{nt})$$ be the set of residual signs at time $$t$$. A generic element $$w_{zu}$$ of the working covariance matrix $$W_i(\eta)$$ can be written as \begin{equation*} w_{zu}= \begin{cases} \tau (1-\tau), & z=u \\ p_{zu} - \tau^2, & z \ne u \\ \end{cases} \end{equation*} where $$p_{zu}=E(V_{z} V_{u})=P(V_{z}=1,V_{u}=1)$$. The elements of $$W_i(\eta)$$ can be computed simultaneously through a second set of estimating equations (Prentice, 1988). These probabilities are bounded by $$\tau$$, the marginal probability of $$V_{z}$$ and $$V_{u}$$, as follows: \begin{equation} \text{max}(0,2 \tau -1) \le p_{zu} \le \tau. \end{equation} (2.3) The correlation coefficient is not a good measure of dependence between binary variables, because it is bounded by their marginal frequencies. Instead, we propose to model a working association matrix defined through odds ratios. Let $$\eta_{zu}$$ be the odds ratio between $$V_{z}$$ and $$V_{u}$$, \begin{equation*} \eta_{zu}=\frac{P(V_{z}=1,V_{u}=1) / P(V_{z}=0,V_{u}=0)}{P(V_{z}=0,V_{u}=1) / P(V_{z}=1,V_{u}=0)}. \end{equation*} Let $$\mathcal{A}=\{(V_z,V_u)\},\;z=u,\dots,T,\; u=1,\dots,T,$$ be the set of pairwise vectors of all $$\dbinom{T}{2}$$ residual signs corresponding to the odds ratios in the lower triangular part of the working correlation matrix $$W_i(\eta)$$. Because $$W_i(\eta)$$ is symmetric, the upper triangular part is equal to the lower part. Consider the new data set $$(V_z, V_u , z, u)$$. For any working structure of $$W_i (\eta)$$, the respective set of odds ratios can be estimated simultaneously through an appropriate definition of the linear predictor in a logistic regression of $$V_z$$ on $$V_u$$, $$V_z|V_u \sim Be(\mu_{zu})$$. Estimating $$W_i(\eta)$$ using logistic models has some advantages. First, it makes it easy to specify the form of the working correlation matrix. For example, Exchangeable: $$\textrm{logit}(\mu_{zu})=\alpha + \eta V_u$$; Toeplitz: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{T-1} \eta_{i} I_{z-u=i} V_u + \sum_{i=1}^{T-1} I_{z-u=i}$$; Unstructured: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{{T}\choose{2}} \eta_{i} I_{c=i} V_u + \sum_{i=1}^{{T}\choose{2}}I_{c=i}$$. Nested Exchangeable: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{K-1} \eta_i V_u I_{c_i=1} + \sum_{i=1}^{K-1} I_{c_i=1}$$, where $$c_i$$ indicates whether $$V_z$$ and $$V_u$$ belong to the same $$i$$-th cluster. Exchangeable varying with a covariate $$X$$: $$\textrm{logit}(\mu_{zu})=\alpha + \sum_{i=1}^{K} \eta_i V_u I_{\textit{X}=i} + \sum_{i=1}^{1} I_{\textit{X}=i}$$, where $$K$$ indicates the number of categories of $$X$$. Second, logistic models can be used to easily select the most appropriate structure of the working correlation matrix. Likelihood comparisons and Akaike’s information criterion may help identify the best structure. Given the marginal probabilities $$\tau$$ and the odds ratios $$\eta_{zu}$$, the joint probabilities $$p_{zu}$$ can be obtained by solving the following equation: \begin{equation*} (\eta_{zu} -1) p_{zu}^2 + ( 2 \tau (1 - \eta_{zu}) -1)p_{zu} + \tau^2 \eta_{zu}=0. \end{equation*} Only the smaller root of the previous equation provides probabilities that respect the constraints in (2.3). 3. Simulation study We conducted simulations to assess the performance of the proposed method. The response variable was generated from the following model: \begin{equation*} y_{ij}=\beta_0 +\beta_1 x_{1ij} + (1+ |x_{1ij}|) (\epsilon_{ij} - q_{\tau}) \;\;\; i=1,\dots,250,\;\;j=1,\dots,T, \end{equation*} where $$q_{\tau}$$ was such that $$p(\epsilon_{ij} \le q_{\tau}) = \tau$$. The size of each cluster was sampled from a binomial distribution $$\textrm{Binom}(T, p)$$, with $$T = (5,10)$$ and $$p = 0.5$$. The covariate $$x_{1ij}$$ was sampled from a uniform distribution $$U(0,1)$$ and the regression coefficients were set to one. The distribution of the error term, $$\epsilon_{i}=(\epsilon_{i1},\dots,\epsilon_{iT})^\top$$, was multivariate Gaussian with variance equal to 10. We considered the following dependence structures: Exchangeable with correlation $$\rho = 0.3$$ and $$\rho = 0.6$$; Independent; Toeplitz with correlation $$\rho = 0.4$$ for the first two lags and zero otherwise. We conducted a simulation study with 500 independent realizations and estimated three different quantiles, $$\tau=(0.10, 0.25, 0.50)$$. All computations were performed using R version 3.23. The programming code is available from the author upon request. The multivariate Gaussian random variables were generated using the “mvtnorm” library. Model-based (naive) standard errors were on average smaller than robust standard errors in all settings (Table 1), except the independence case. Misspecification of the working correlation structure lead to a decrease in the empirical coverage of the estimators. Coverage probabilities of the robust standard errors were close to their nominal value in all the settings (Table 2). The estimator of the 10-th percentile showed undercoverage, with smallest observed value equal to 0.88. The average standard errors of the estimators were near those observed over the Monte Carlo replicates across all settings (Tables 3 and 4). Cluster bootstrap had a smaller mean squared error (MSE) and average standard error than the proposed method when the 10-th percentile was estimated (Table 5). When the 25-th and 50-th percentile were estimated, the proposed method provided smaller MSE and average standard error than cluster bootstrap, regardless of the dependence structure. Similarly to the classic GEE for the expectation of a response variable, selecting the true dependence structure improved the performances of the estimator. When the true dependence structure was Toeplitz, the estimator obtained using the true dependence structure provided the smallest MSE and average standard errors in most of the settings. In the independent case, results were very similar across all the working structures. In the exchangeable case, the independence and exchangeable structures had the smallest MSE and average standard errors. We computed also the bias of the estimator, which is shown in Table 6. Table 1 Standard error difference between the robust and naive estimators $$($$Avg s.e.$$)$$ and empirical coverage of the naive estimators $$($$Cover$$)$$ for the parameter $$\beta_1$$ at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; TO, Toeplitz $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 Table 1 Standard error difference between the robust and naive estimators $$($$Avg s.e.$$)$$ and empirical coverage of the naive estimators $$($$Cover$$)$$ for the parameter $$\beta_1$$ at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; TO, Toeplitz $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ Avg s.e. Cover Avg s.e. Cover Avg s.e. Cover True structure: Toeplitz $$\quad$$ WI 0.2036 0.796 0.1260 0.868 0.1217 0.864 $$\quad$$ EX 10 0.0965 0.856 0.0090 0.946 0.0067 0.922 $$\quad$$ TO 0.1039 0.848 0.0216 0.940 0.0209 0.936 $$\quad$$ WI 0.2932 0.750 0.1479 0.880 0.1399 0.890 $$\quad$$ EX 5 0.1667 0.786 0.0161 0.926 0.0035 0.940 $$\quad$$ TO 0.1694 0.792 0.0229 0.918 0.0124 0.938 True structure: Independence $$\quad$$ WI 0.0418 0.860 –0.0197 0.954 –0.0161 0.956 $$\quad$$ EX 10 0.0447 0.860 –0.0171 0.952 –0.0136 0.950 $$\quad$$ TO 0.0447 0.862 –0.0171 0.954 –0.0133 0.950 $$\quad$$ WI 0.1038 0.854 –0.0184 0.924 –0.0256 0.966 $$\quad$$ EX 5 0.1076 0.848 –0.0147 0.922 –0.0230 0.964 $$\quad$$ TO 0.1078 0.844 –0.0141 0.924 –0.0228 0.960 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.2887 0.776 0.2298 0.830 0.2274 0.800 $$\quad$$ EX 10 0.1173 0.844 0.0554 0.942 0.0570 0.936 $$\quad$$ TO 0.1184 0.846 0.0552 0.944 0.0571 0.938 $$\quad$$ WI 0.2715 0.774 0.1478 0.890 0.1473 0.866 $$\quad$$ EX 5 0.1541 0.804 0.0100 0.934 0.0079 0.946 $$\quad$$ TO 0.1564 0.804 0.0118 0.938 0.0078 0.942 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.5468 0.658 0.4402 0.714 0.4234 0.680 $$\quad$$ EX 10 0.2913 0.814 0.2358 0.856 0.2388 0.828 $$\quad$$ TO 0.2918 0.810 0.2373 0.848 0.2416 0.836 $$\quad$$ WI 0.5100 0.628 0.3237 0.786 0.2870 0.810 $$\quad$$ EX 5 0.2908 0.752 0.1190 0.904 0.0998 0.912 $$\quad$$ TO 0.2928 0.740 0.1216 0.904 0.1024 0.908 Table 2 Empirical coverage of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 Table 2 Empirical coverage of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.914 0.926 0.934 0.950 0.940 0.938 $$\quad$$ EX 0.914 0.926 0.930 0.948 0.938 0.934 $$\quad$$ AR 10 0.912 0.920 0.930 0.946 0.944 0.934 $$\quad$$ TO 0.924 0.922 0.924 0.948 0.938 0.940 $$\quad$$ BS 0.944 0.940 0.946 0.956 0.938 0.942 $$\quad$$ WI 0.910 0.902 0.926 0.946 0.944 0.944 $$\quad$$ EX 0.912 0.902 0.926 0.948 0.948 0.940 $$\quad$$ AR 5 0.912 0.896 0.924 0.950 0.944 0.940 $$\quad$$ TO 0.912 0.906 0.914 0.940 0.938 0.932 $$\quad$$ BS 0.934 0.926 0.938 0.948 0.958 0.950 True structure: Independence $$\quad$$ WI 0.914 0.928 0.952 0.950 0.942 0.948 $$\quad$$ EX 0.910 0.928 0.950 0.950 0.942 0.942 $$\quad$$ AR 10 0.914 0.926 0.950 0.950 0.940 0.948 $$\quad$$ TO 0.906 0.932 0.948 0.944 0.940 0.946 $$\quad$$ BS 0.938 0.958 0.966 0.966 0.948 0.954 $$\quad$$ WI 0.908 0.922 0.942 0.926 0.946 0.956 $$\quad$$ EX 0.908 0.920 0.946 0.928 0.946 0.954 $$\quad$$ AR 5 0.910 0.920 0.946 0.926 0.946 0.954 $$\quad$$ TO 0.912 0.918 0.942 0.932 0.946 0.950 $$\quad$$ BS 0.930 0.930 0.958 0.944 0.964 0.964 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.926 0.930 0.934 0.954 0.946 0.948 $$\quad$$ EX 0.934 0.930 0.932 0.962 0.948 0.950 $$\quad$$ AR 10 0.932 0.932 0.932 0.954 0.950 0.954 $$\quad$$ TO 0.928 0.934 0.934 0.960 0.946 0.950 $$\quad$$ BS 0.942 0.950 0.940 0.958 0.964 0.956 $$\quad$$ WI 0.892 0.906 0.946 0.944 0.922 0.926 $$\quad$$ EX 0.888 0.900 0.948 0.946 0.924 0.938 $$\quad$$ AR 5 0.894 0.898 0.950 0.946 0.920 0.938 $$\quad$$ TO 0.892 0.904 0.948 0.944 0.918 0.938 $$\quad$$ BS 0.942 0.954 0.950 0.958 0.940 0.938 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.902 0.908 0.954 0.936 0.956 0.942 $$\quad$$ EX 0.900 0.914 0.944 0.930 0.944 0.944 $$\quad$$ AR 10 0.908 0.908 0.946 0.926 0.944 0.944 $$\quad$$ TO 0.904 0.916 0.944 0.930 0.942 0.946 $$\quad$$ BS 0.932 0.948 0.962 0.942 0.952 0.948 $$\quad$$ WI 0.898 0.922 0.938 0.946 0.932 0.934 $$\quad$$ EX 0.906 0.910 0.944 0.946 0.938 0.946 $$\quad$$ AR 5 0.912 0.908 0.934 0.944 0.934 0.946 $$\quad$$ TO 0.902 0.908 0.928 0.950 0.934 0.944 $$\quad$$ BS 0.938 0.946 0.966 0.968 0.956 0.946 Table 3 Average standard error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 Table 3 Average standard error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.360 0.757 0.289 0.610 0.270 0.571 $$\quad$$ EX 0.360 0.756 0.288 0.610 0.270 0.570 $$\quad$$ AR 10 0.358 0.755 0.287 0.605 0.268 0.566 $$\quad$$ TO 0.356 0.750 0.285 0.601 0.266 0.562 $$\quad$$ BS 0.369 0.765 0.305 0.642 0.283 0.599 $$\quad$$ WI 0.492 1.066 0.389 0.824 0.372 0.778 $$\quad$$ EX 0.491 1.062 0.388 0.823 0.371 0.775 $$\quad$$ AR 5 0.490 1.059 0.385 0.605 0.365 0.770 $$\quad$$ TO 0.488 1.053 0.382 0.813 0.363 0.763 $$\quad$$ BS 0.503 1.063 0.417 0.882 0.396 0.827 True structure: Independence $$\quad$$ WI 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ EX 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ AR 10 0.288 0.611 0.219 0.464 0.205 0.435 $$\quad$$ TO 0.287 0.609 0.219 0.463 0.205 0.434 $$\quad$$ BS 0.302 0.630 0.235 0.496 0.218 0.460 $$\quad$$ WI 0.408 0.880 0.318 0.666 0.292 0.615 $$\quad$$ EX 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ AR 5 0.408 0.880 0.319 0.666 0.292 0.615 $$\quad$$ TO 0.407 0.878 0.318 0.665 0.291 0.614 $$\quad$$ BS 0.426 0.902 0.346 0.722 0.316 0.663 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.404 0.862 0.338 0.715 0.323 0.677 $$\quad$$ EX 0.404 0.863 0.337 0.712 0.323 0.676 $$\quad$$ AR 10 0.403 0.862 0.337 0.713 0.322 0.678 $$\quad$$ TO 0.403 0.860 0.336 0.714 0.322 0.676 $$\quad$$ BS 0.414 0.877 0.353 0.748 0.337 0.709 $$\quad$$ WI 0.483 1.041 0.396 0.827 0.373 0.782 $$\quad$$ EX 0.483 1.042 0.394 0.824 0.372 0.780 $$\quad$$ AR 5 0.483 1.043 0.395 0.825 0.372 0.780 $$\quad$$ TO 0.484 1.041 0.394 0.824 0.371 0.778 $$\quad$$ BS 0.503 1.063 0.423 0.885 0.399 0.836 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.533 1.124 0.449 0.933 0.421 0.883 $$\quad$$ EX 0.533 1.125 0.443 0.922 0.416 0.870 $$\quad$$ AR 10 0.533 1.123 0.445 0.930 0.418 0.882 $$\quad$$ TO 0.530 1.116 0.449 0.927 0.420 0.876 $$\quad$$ BS 0.541 1.125 0.461 0.957 0.432 0.905 $$\quad$$ WI 0.604 1.280 0.472 1.000 0.441 0.921 $$\quad$$ EX 0.598 1.275 0.471 0.997 0.440 0.923 $$\quad$$ AR 5 0.603 1.279 0.471 1.000 0.441 0.923 $$\quad$$ TO 0.594 1.270 0.470 0.996 0.440 0.924 $$\quad$$ BS 0.612 1.272 0.501 1.060 0.467 0.974 Table 4 Monte Carlo standard deviation of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 Table 4 Monte Carlo standard deviation of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.371 0.810 0.301 0.616 0.282 0.583 $$\quad$$ EX 0.371 0.810 0.301 0.617 0.281 0.584 $$\quad$$ AR 10 0.370 0.807 0.299 0.612 0.280 0.579 $$\quad$$ TO 0.370 0.806 0.299 0.603 0.279 0.573 $$\quad$$ BS 0.356 0.772 0.300 0.615 0.286 0.592 $$\quad$$ WI 0.536 1.141 0.413 0.858 0.376 0.805 $$\quad$$ EX 0.534 1.136 0.411 0.850 0.371 0.792 $$\quad$$ AR 5 0.530 1.132 0.410 0.851 0.374 0.792 $$\quad$$ TO 0.529 1.128 0.407 0.849 0.371 0.790 $$\quad$$ BS 0.504 1.048 0.410 0.856 0.382 0.817 True structure: Independence $$\quad$$ WI 0.314 0.660 0.217 0.452 0.212 0.426 $$\quad$$ EX 0.314 0.661 0.217 0.453 0.212 0.426 $$\quad$$ AR 10 0.315 0.661 0.217 0.455 0.212 0.426 $$\quad$$ TO 0.314 0.663 0.218 0.453 0.212 0.427 $$\quad$$ BS 0.296 0.613 0.217 0.451 0.214 0.430 $$\quad$$ WI 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ EX 0.433 0.920 0.314 0.686 0.276 0.583 $$\quad$$ AR 5 0.435 0.922 0.314 0.685 0.276 0.583 $$\quad$$ TO 0.433 0.923 0.315 0.685 0.275 0.583 $$\quad$$ BS 0.414 0.873 0.316 0.685 0.284 0.606 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.421 0.893 0.340 0.688 0.333 0.676 $$\quad$$ EX 0.420 0.896 0.339 0.684 0.333 0.671 $$\quad$$ AR 10 0.421 0.895 0.339 0.687 0.333 0.668 $$\quad$$ TO 0.421 0.894 0.340 0.684 0.331 0.668 $$\quad$$ BS 0.409 0.859 0.348 0.705 0.340 0.693 $$\quad$$ WI 0.539 1.174 0.383 0.804 0.391 0.806 $$\quad$$ EX 0.538 1.169 0.381 0.796 0.387 0.793 $$\quad$$ AR 5 0.540 1.176 0.382 0.798 0.390 0.795 $$\quad$$ TO 0.537 1.162 0.382 0.794 0.389 0.793 $$\quad$$ BS 0.473 1.000 0.386 0.803 0.390 0.808 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.564 1.181 0.433 0.933 0.422 0.907 $$\quad$$ EX 0.558 1.197 0.437 0.941 0.421 0.874 $$\quad$$ AR 10 0.561 1.199 0.434 0.949 0.429 0.902 $$\quad$$ TO 0.562 1.197 0.438 0.950 0.426 0.885 $$\quad$$ BS 0.522 1.094 0.436 0.940 0.425 0.884 $$\quad$$ WI 0.656 1.414 0.482 1.021 0.431 0.944 $$\quad$$ EX 0.644 1.388 0.476 0.998 0.431 0.934 $$\quad$$ AR 5 0.650 1.407 0.486 1.024 0.431 0.930 $$\quad$$ TO 0.652 1.403 0.481 1.019 0.431 0.933 $$\quad$$ BS 0.598 1.224 0.469 0.982 0.433 0.943 Table 5 Mean squared error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 Table 5 Mean squared error of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.1381 0.6603 0.0903 0.3789 0.0798 0.3414 $$\quad$$ EX 0.1387 0.6593 0.0907 0.3802 0.0796 0.3430 $$\quad$$ AR 10 0.1373 0.6555 0.0889 0.3737 0.0790 0.3365 $$\quad$$ TO 0.1366 0.6522 0.0890 0.3633 0.0783 0.3286 $$\quad$$ BS 0.1273 0.5966 0.0899 0.3772 0.0820 0.3514 $$\quad$$ WI 0.2880 1.3025 0.1699 0.7360 0.1410 0.6469 $$\quad$$ EX 0.2863 1.2933 0.1685 0.7240 0.1373 0.6269 $$\quad$$ AR 5 0.2825 1.2753 0.1678 0.7216 0.1372 0.6259 $$\quad$$ TO 0.2818 1.2864 0.1656 0.7259 0.1399 0.6226 $$\quad$$ BS 0.2536 1.1050 0.1675 0.7340 0.1455 0.6673 True structure: Independence $$\quad$$ WI 0.0992 0.4368 0.0480 0.2068 0.0449 0.1811 $$\quad$$ EX 0.0992 0.4381 0.0481 0.2074 0.0448 0.1810 $$\quad$$ AR 10 0.0992 0.4377 0.0481 0.2074 0.0449 0.1810 $$\quad$$ TO 0.0997 0.4396 0.0487 0.2098 0.0448 0.1820 $$\quad$$ BS 0.0881 0.3795 0.0482 0.2054 0.0459 0.1847 $$\quad$$ WI 0.1868 0.8453 0.0984 0.4693 0.0760 0.3392 $$\quad$$ EX 0.1869 0.8462 0.0984 0.4694 0.0758 0.3387 $$\quad$$ AR 5 0.1870 0.8485 0.0984 0.4691 0.0758 0.3386 $$\quad$$ TO 0.1892 0.8507 0.0989 0.4689 0.0758 0.3394 $$\quad$$ BS 0.1711 0.7612 0.0995 0.4678 0.0808 0.3660 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI 0.1771 0.7963 0.1160 0.4746 0.1106 0.4563 $$\quad$$ EX 0.1761 0.8009 0.1152 0.4702 0.1105 0.4490 $$\quad$$ AR 10 0.1772 0.7989 0.1151 0.4697 0.1095 0.4457 $$\quad$$ TO 0.1772 0.7977 0.1157 0.4739 0.1105 0.4456 $$\quad$$ BS 0.1667 0.7360 0.1212 0.4990 0.1156 0.4804 $$\quad$$ WI 0.2905 1.3773 0.1462 0.6456 0.1526 0.6486 $$\quad$$ EX 0.2886 1.3648 0.1454 0.6326 0.1500 0.6286 $$\quad$$ AR 5 0.2913 1.3820 0.1456 0.6357 0.1517 0.6311 $$\quad$$ TO 0.2875 1.3484 0.1461 0.6290 0.1509 0.6283 $$\quad$$ BS 0.2242 0.9985 0.1492 0.6439 0.1519 0.6523 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.3107 1.3797 0.1883 0.8730 0.1783 0.7632 $$\quad$$ EX 0.3173 1.4302 0.1909 0.9024 0.1834 0.8213 $$\quad$$ AR 10 0.3145 1.3921 0.1890 0.8874 0.1774 0.7827 $$\quad$$ TO 0.3153 1.4352 0.1923 0.9044 0.1813 0.8117 $$\quad$$ BS 0.2730 1.1983 0.1903 0.8910 0.1805 0.7805 $$\quad$$ WI 0.4306 1.9953 0.2268 0.9978 0.1858 0.8891 $$\quad$$ EX 0.4146 1.9234 0.2323 1.0414 0.1857 0.8707 $$\quad$$ AR 5 0.4219 1.9758 0.2317 1.0377 0.1859 0.8647 $$\quad$$ TO 0.4250 1.9645 0.2360 1.0470 0.1854 0.8701 $$\quad$$ BS 0.3570 1.4985 0.2210 0.9626 0.1869 0.8874 Table 6 Bias of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 Table 6 Bias of the proposed estimator at three quantiles for different working correlation structures, cluster size $$T$$ and true correlation structures. WI, independence; EX, exchangeable; AR, autoregressive; TO, Toeplitz; BS, cluster bootstrap $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ $$T$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ $$\hat{\beta}_0$$ $$\hat{\beta}_1$$ True structure: Toeplitz $$\quad$$ WI 0.0231 –0.0750 –0.0021 0.0103 –0.0270 0.0493 $$\quad$$ EX 0.0218 –0.0718 –0.0011 0.0101 –0.0288 0.0513 $$\quad$$ AR 10 0.0225 –0.0732 –0.0009 0.0081 –0.0266 0.0466 $$\quad$$ TO 0.0189 –0.0678 0.0023 0.0010 –0.0220 0.0341 $$\quad$$ BS 0.0272 –0.0419 0.0032 0.0133 –0.0247 0.0459 $$\quad$$ WI –0.0396 0.0575 –0.0086 0.0406 0.0029 –0.0176 $$\quad$$ EX –0.0407 0.0663 –0.0107 0.0499 0.0029 –0.0119 $$\quad$$ AR 5 –0.0431 0.0712 –0.0085 0.0429 0.0027 –0.0114 $$\quad$$ TO –0.0528 0.0889 –0.0131 0.0515 0.0005 –0.0039 $$\quad$$ BS –0.0203 0.0991 –0.0025 –0.0555 0.0101 –0.0326 True structure: Independence $$\quad$$ WI –0.0247 0.0379 0.0314 –0.0509 –0.0026 –0.0122 $$\quad$$ EX –0.0245 0.0374 0.0315 –0.0512 –0.0026 –0.0121 $$\quad$$ AR 5 –0.0247 0.0378 0.0314 –0.0511 –0.0025 –0.0122 $$\quad$$ TO –0.0236 0.0363 0.0328 –0.0533 –0.0027 –0.0125 $$\quad$$ BS –0.047 0.0644 0.0335 –0.0476 –0.0027 –0.0129 $$\quad$$ WI –0.0033 –0.0253 –0.0042 0.0137 0.0043 0.0003 $$\quad$$ EX –0.0033 –0.0210 –0.0044 0.0141 0.0045 0.0002 $$\quad$$ AR 5 –0.0032 –0.0257 –0.0047 0.0148 0.0043 0.0005 $$\quad$$ TO –0.0044 –0.0210 –0.0046 0.0150 0.0044 0.0008 $$\quad$$ BS 0.0101 0.0003 0.0019 0.0143 0.0064 –0.0043 True structure: Exchangeable with $$\rho=0.3$$ $$\quad$$ WI –0.0201 0.0023 –0.0208 0.0521 –0.0007 –0.0188 $$\quad$$ EX –0.0192 –0.0029 0.0212 0.0540 –0.0023 –0.0181 $$\quad$$ AR 5 –0.0179 –0.0032 –0.0204 0.0526 –0.0015 –0.0164 $$\quad$$ TO –0.0188 –0.0007 –0.0216 0.0538 –0.0026 –0.0169 $$\quad$$ BS 0.0005 0.0164 –0.0116 0.0531 –0.0003 –0.0216 $$\quad$$ WI 0.0032 –0.0504 0.0096 –0.0004 –0.0077 –0.0133 $$\quad$$ EX 0.0010 –0.0429 0.0142 –0.0045 0.0115 –0.0166 $$\quad$$ AR 5 0.0002 –0.0430 0.0120 –0.0018 0.0106 –0.0171 $$\quad$$ TO –0.0010 –0.0353 –0.0140 –0.0040 0.0123 –0.0181 $$\quad$$ BS 0.0343 –0.0252 0.0206 0.0012 0.0012 –0.0073 True structure: Exchangeable with $$\rho=0.6$$ $$\quad$$ WI 0.0053 0.00171 –0.0312 0.0697 –0.0141 –0.0099 $$\quad$$ EX 0.0148 0.0060 –0.0250 0.0620 –0.0104 –0.0254 $$\quad$$ AR 10 0.0072 0.0164 –0.0289 0.0601 –0.0144 –0.0162 $$\quad$$ TO 0.0138 0.0088 –0.0252 0.0606 –0.0115 –0.0210 $$\quad$$ BS 0.0282 0.0646 –0.0270 0.0944 –0.0159 –0.0034 $$\quad$$ WI –0.0233 –0.0092 0.0328 –0.0550 0.0022 –0.0167 $$\quad$$ EX –0.0163 –0.0212 0.0197 –0.0360 0.0082 –0.0239 $$\quad$$ AR 5 –0.0178 –0.0251 0.0212 –0.0378 0.0098 –0.0282 $$\quad$$ TO –0.0157 –0.0190 0.0203 –0.0371 0.0072 –0.0208 $$\quad$$ BS 0.0086 0.0463 0.0318 –0.0137 0.0038 –0.0211 Model selection performed through Akaike’s information criterion selected the true dependence structure with high probability (Table 7). The smallest percentage of correct selection was 40%, obtained when the true dependence structure was Toeplitz, $$\tau=0.10$$ and $$T=5$$. The highest was 100%, obtained when the true dependence structure was Toeplitz, $$\tau=0.50$$ and $$T=10$$. In general, the percentage of correct selection was higher for the median than for the 10-th and 25-th percentiles. Table 7 Observed percentage of correct selections of the true model in different scenarios and quantiles. $$T$$ indicates the cluster size. WI, independence; EX, exchangeable; TO, Toeplitz; AR, autoregressive $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 Table 7 Observed percentage of correct selections of the true model in different scenarios and quantiles. $$T$$ indicates the cluster size. WI, independence; EX, exchangeable; TO, Toeplitz; AR, autoregressive $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 $$\tau=0.10$$ $$\tau=0.25$$ $$\tau=0.50$$ WI EX TO AR WI EX TO AR WI EX TO AR True model $$T$$ (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Independent 10 71 17% 12 0 79 17 4 0 83 17 0 0 5 74 17% 11 0 79 17 5 0 83 15 2 0 Exchangeable 10 2 81% 16 0 0 88 12 0 0 100 0 0 with $$\rho = 0.3$$ 5 6 81% 13 0 1 90 9 0 0 99 1 0 Exchangeable 10 0 73% 27 0 0 81 19 0 0 95 5 0 with $$\rho = 0.6$$ 5 0 81% 19 0 0 89 11 0 0 92 8 0 Toeplitz 10 6 29% 65 0 0 8 92 0 0 0 100 0 5 5 56% 40 0 0 28 71 0 0 3 97 0 We also explored the effect of cluster size by raising the binomial probability in the data-generating model to $$p = 0.8$$. The results were nearly unchanged and are not shown. We also considered a sample size of $$n = 100$$ and $$T = 15$$. The properties of the estimators were similar to those obtained with the previous settings. (The results are not shown.) 4. Application to internet-based cognitive therapy data set The data set consisted of 95 ($$n=95$$) families with a child aged 8–12 years with a principal diagnosis of generalized anxiety, panic disorder, separation anxiety, social phobia, and specific phobia. Participants were randomized to ten weeks of internet-based cognitive therapy with therapist support ($$70$$ families), or to a waitlist control condition ($$25$$ families). At weekly intervals, the amount of difficulties in emotion regulation scale (DERS) was measured on a 0–100 scale, where 0 indicated no difficulties and 100 extreme difficulties. The maximum number of repeated measurements is fourteen ($$T=14$$). Similarly to previous studies, we assumed that data were missing completely at random. The main interest of the study was to assess the relationship between the treatment and DERS. Figure 1 shows the residual errors of a classical mean regression and their kernel density. It was markedly skewed and there were several outliers. The mean was not an appropriate summary of it, and we considered quantiles. There seemed to be a time effect on the response variable, both in treated and untreated groups (Figure 2). The intraclass correlation was estimated to be 70%. Fig. 1. View largeDownload slide Mean regression residuals obtained from model on the right-hand side of the equation (4.4) and their estimated kernel density. Fig. 1. View largeDownload slide Mean regression residuals obtained from model on the right-hand side of the equation (4.4) and their estimated kernel density. Fig. 2. View largeDownload slide Boxplot of DERS at different times in treated (top panel) and untreated (bottom panel) children. In the treated group the sample sizes at the 14 time points were, respectively: 58, 52, 51, 41, 49, 39, 43, 39, 43, 35, 27, 32, 32, and 30 children. In the untreated group the sample sizes at the 14 time points were, respectively: 19, 20, 16, 16, 13, 14, 13, 12, 13, 9, 12, 12, 12, and 12 children. Fig. 2. View largeDownload slide Boxplot of DERS at different times in treated (top panel) and untreated (bottom panel) children. In the treated group the sample sizes at the 14 time points were, respectively: 58, 52, 51, 41, 49, 39, 43, 39, 43, 35, 27, 32, 32, and 30 children. In the untreated group the sample sizes at the 14 time points were, respectively: 19, 20, 16, 16, 13, 14, 13, 12, 13, 9, 12, 12, 12, and 12 children. We considered three different quantiles, $$\tau=(0.25, 0.50, 0.75)$$, and estimated the following model: \begin{equation} Q(\textrm{Ders}_{ij} | \textrm{Treat}, \textrm{Time})=\beta_0 +\beta_1 \textrm{Treat}_{ij} + \beta_2 \textrm{Time}_{ij} + \beta_3 \textrm{Treat}_{ij} \times \textrm{Time}_{ij}. \end{equation} (4.4) We considered an exchangeable, Toeplitz, independence, and exchangeable varying with treatment correlation structures, along with a cluster bootstrap. Results are shown in Table 8. Table 8 Estimated quantile regression coefficients, standard errors $$($$in parentheses$$)$$ and logistic models’ AIC for three quantiles using exchangeable $$($$EX$$)$$, exchangeable varying with treatment $$($$EX-Trt$$)$$, Toeplitz $$($$TO$$)$$, and independence $$($$WI$$)$$ working correlation structures, along with a cluster bootstrap $$($$BT$$)$$ Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) Table 8 Estimated quantile regression coefficients, standard errors $$($$in parentheses$$)$$ and logistic models’ AIC for three quantiles using exchangeable $$($$EX$$)$$, exchangeable varying with treatment $$($$EX-Trt$$)$$, Toeplitz $$($$TO$$)$$, and independence $$($$WI$$)$$ working correlation structures, along with a cluster bootstrap $$($$BT$$)$$ Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) Coefficient $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ EX Intercept 53.208* (0.908) 65.273* (3.243) 72.915* (2.739) Treatment $$-$$5.988* (2.589) $$-$$8.320* (3.908) $$-$$6.406 (3.790) Time $$-$$0.630 (0.648) $$-$$0.608* (0.270) $$-$$0.446* (0.218) TrxTime 0.202 (0.694) 0.081 (0.307) 0.265 (0.301) AIC 3165 4043 3045 EX-Trt Intercept 53.152* (0.936) 65.282* (3.250) 72.933* (2.745) Treatment $$-$$5.766* (2.576) $$-$$8.221* (3.894) $$-$$6.336 (3.728) Time $$-$$0.649 (0.684) $$-$$0.607* (0.270) $$-$$0.445* (0.219) TrxTime 0.227 (0.728) 0.081 (0.307) 0.265 (0.300) AIC 3092 4031 3022 TO Intercept 53.436* (0.611) 63.528* (6.809) 72.068* (2.467) Treatment $$-$$6.964* (1.771) $$-$$7.140 (6.950) $$-$$6.307 (3.454) Time $$-$$0.810 (0.658) $$-$$0.472 (0.445) $$-$$0.434* (0.198) Tr$$\times$$Time 0.579 (0.689) 0.004 (0.478) 0.290 (0.300) AIC 3179 4037 3062 WI Intercept 53.993* (3.392) 64.972* (3.560) 72.539* (2.639) Treatment $$-$$6.079 (3.937) $$-$$6.954 (4.097) $$-$$5.240 (3.489) Time $$-$$0.542 (0.426) $$-$$0.679* (0.288) $$-$$0.534* (0.214) TrxTime 0.120 (0.486) 0.211 (0.348) 0.319 (0.313) AIC 4207 5177 4207 BT Intercept 54.430* (3.161) 65.059* (4.104) 72.229* (2.940) Treatment $$-$$5.180 (3.854) $$-$$6.171 (4.383) $$-$$4.629 (3.914) Time $$-$$0.503 (0.322) $$-$$0.728* (0.366) $$-$$0.627* (0.254) TrxTime $$-$$0.080 (0.406) 0.187 (0.417) 0.391 (0.345) We found a significant effect of treatment at the $$50$$-th and $$25$$-th percentiles in the exchangeable and exchangeable varying with treatment correlation structures. In the Toeplitz structure, this effect was significant only at the $$25$$-th percentile. The coefficient for time was significant at the median in all correlation structures except the Toeplitz. It was significant at the $$75$$-th percentile in all structures. The exchangeable structure varying with treatment had the smallest Akaike’s information criterion. Measurements from subjects that were treated were less correlated than those from subjects that were not treated (Table 9). Standard errors obtained with cluster bootstrap were up to 50% greater than those obtained with the proposed method. Table 9 Estimated working correlation parameters for the exchangeable ($$\phi_{\textrm{Exch}}$$) and exchangeable varying with treatment ($$\phi_{\textrm{TRT=1}}$$ and $$\phi_{\textrm{TRT=0}}$$) structures Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Table 9 Estimated working correlation parameters for the exchangeable ($$\phi_{\textrm{Exch}}$$) and exchangeable varying with treatment ($$\phi_{\textrm{TRT=1}}$$ and $$\phi_{\textrm{TRT=0}}$$) structures Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Correlation parameter $$\tau=0.25$$ $$\tau=0.50$$ $$\tau=0.75$$ $$\phi_{\textrm{TRT=0}}$$ 0.06 0.11 0.08 $$\phi_{\textrm{TRT=1}}$$ 0.12 0.14 0.12 $$\phi_{\textrm{Exch}}$$ 0.10 0.13 0.11 Figure 3 shows the relationship between DERS and time, stratified by treatment. As the estimated quantile increases, the correlation between DERS and time decreases. The differences between treated and untreated are stronger at the median than at other percentiles. Fig. 3. View largeDownload slide Quantile regression for DERS as a function of time by treatment at three different quantiles $$\tau=(0.25,0.50,0.75)$$ with an exchangeable correlation structure varying with treatment. The dashed lines represent the treated group, while the solid lines represent the untreated group. Fig. 3. View largeDownload slide Quantile regression for DERS as a function of time by treatment at three different quantiles $$\tau=(0.25,0.50,0.75)$$ with an exchangeable correlation structure varying with treatment. The dashed lines represent the treated group, while the solid lines represent the untreated group. Because the interaction term was not significant in all percentiles, we concluded that the treatment did not significantly decrease the anxiety disorder of children. 5. Discussion This article describes an alternative method to estimate the dependence within clusters for inference on population marginal quantile models. The association matrix of the residuals’ sign is estimated by means of appropriate logistic regression models. The estimation is flexible and computationally fast. In our study, the proposed method was approximately as fast as GEE applied to the mean. At extreme quantiles, the algorithm occasionally failed to converge. This partly accounts for the observed larger MSE of the proposed method with respect to the bootstrap at the 10-percentile in our simulation study. The best-fitting working structure can be selected by likelihood comparisons or, for non-nested models, by the Akaike’s information criterion. Our simulations showed that this criterion could select the true dependence structure with high probability. In other settings, however, other criteria may be preferable (Jaman and others, 2016). In the scenarios considered in our simulation, the asymptotic approximations for the covariance matrix of the estimators were satisfactory. When the asymptotic approximations are suspected to be poor, one may consider running a simulation specifically tailored to the data at hand. Possible extensions of this work include models for dependence with non-parametric quantile regression, M-estimators, LMS (Cole, 1990), and other classifiers. When missing data are present, and the non-response mechanism is suspected to be missing at random, weighted estimating equations can be applied. Yi and He (2009) discussed the use of weighted estimating equations for median regression for longitudinal data with dropouts, and demonstrated the consistency and asymptotic distribution of the resulting estimators. Acknowledgments onflict of Interest: None declared. References Brown B. M. and Wang Y.-G. ( 2005 ). Standard errors and covariance matrices for smoothed rank estimators. 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Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Journal

BiostatisticsOxford University Press

Published: Oct 1, 2018

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